Briefly, I'd say, "yes" to your question. Even for more relatable questions, the students still draw the same sort of invalid answer.
I find it a stretch that students can't understand the shepherd problem. If they can't relate to it, then we've done a very poor job of educating them about the world, history, and a lot of other things which have very little to do with mathematics.
Nevertheless, I don't fully agree with the analysis of the linked article either. Ask the students a completely different question, "There are 125 sheep and 5 dogs in a flock. What is the shepherd's name?" Whether or not 75% will give you an answer, you will still see a disappointingly large percentage of students providing you an answer.
Student, "The shepherd's name is John."
Teacher, "How did you come to that conclusion?"
Student, "Well, it could be."
Let's just ignore that the student didn't answer the teacher's question. I've encountered this scenario numerous times in the classroom where a question doesn't really have an answer, so the students make something up, and, because the answer has some kind of plausibility to it, the students think they're done.
The way I would work with these sorts of problems is to pepper them throughout the course. But also, I would focus on the reasoning skills the students need to solve the problem. So, you can have a discussion of what's sorts of information you need to determine someone's age. This can lead to all sorts of interesting answers. For example, many students will offer, "their birthday." The students can relatively quickly figure out that knowing a birthday is insufficient; eventually they'll come up with, "their birth date" or something equivalent. And then the teacher can ask, "do you need to know their birth date to determine their age?" This becomes more challenging for them because the previous answer has locked many of their minds into one way of thinking about the problem. Yet, with some patience and good give-and-take between the teacher and the students, they should be able to see that you can answer such questions as, "My sister was born when my mom was 30 years old. My mom is now 45. How old is my sister?"
I am personally fond of questions like "Jackie's brother is five years older than Paul. Paul was born in 2002. How old is Jackie's brother?" This isn't quite like the shepherd problem, but it's similar in that there isn't quite enough information to know with 100% certainty. However, there are two plausible answers.
So back to the teacher's question, "how did you come to that conclusion?"
The students need to be taught how to explain how they're getting their answers. In an English class, saying something like, "It could be John", might be perfectly acceptable (I'm not an English teacher so I won't weigh in too strongly about the validity of that answer). But, in mathematics and often in the sciences, a plausible answer is usually not what we're looking for. We're usually looking for reasoning that starts for a collection of facts and draws various conclusions based upon those facts.
So, it might be plausible that the shepherd is 25 years old. It might just happen that way. Great. But is this supported by the facts? I would approach this as outlined above: by stepping back and having the students review what sort of reasoning skills they need to use and how fact relate to the sorts of answers they're trying to come up with. And then once you've perhaps convinced them that we can't answer a question about the shepherd's age, ask the students, "what sorts of questions could we ask about this shepherd from the facts presented?"
Ultimately, the students should be able to answer a question like, "There are 125 gabbalahs and 5 tumps. What is Margaret's age?" If the students give up and say, "I don't know. What's a gabbalah? I don't know what a tump is", then the students are again missing the point of the question. Giving up isn't the same thing as saying the question as presented isn't answerable. In fact, what I'd most want to hear from the students is, "Do gabbalah and tumps have anything to do with figuring out someone's age?" That's a great question.
The younger the student, generally, the more a concrete thinker the student is. It's very challenging for them to think about something abstractly (such as in the gabbalah/tump question). But also, I would say that it's not a failure so much of mathematics education that makes the students think they need to come up with an answer. It's about how we teach the students over all. Our whole system of education is biased toward asking questions that the students can answer. (It's not too hard to see why we do that.) And, this bias in the pedagogy biases the students in how they approach problems.
While I may disagree that the shepherd question is fusty, even if it is, that should not be preventing the students from answering the question. In our pedagogical aim to make problem relevant to students, we lose sight that we should also be expanding their horizons, broadening their perspectives, and showing and teaching them ways of grappling with the unfamiliar. (Too often we're encouraging them to stay within their myopic world views by only presenting what is already known and familiar.) Speaking as a mathematician or a scientist, there is very little of interest done in mathematics (or the sciences) which is commonplace; what is interesting in mathematics and the sciences are those places where we lack knowledge and understanding---where our comfort zone is being challenged.